The definition of a cupola, according to my Random House College
Dictionary reads as follows:

cu·po·la(kyou' puh luh), noun1.Archit.a. A light structure on
a dome or roof, serving as a belfry, lantern, or belvedere. b.
a dome, esp. one covering a circular or polygonal area. 2.
any of various domelike structures. 3.Foundry.
a vertical furnace for melting iron to be cast. [< It < L cupula,
see CUP (from Latin, meaning tub or cask) + ULE. (from Latin,
meaning small)]

This site will concern itself with the architectural meanings of the
word. My Penguin Dictionary of Architecture simply
describes a cupola this way:

cu·po·la a dome, especially a
small dome on a circular or polygonal base crowning a roof or turret.

Leaning toward the Random House1b
definition, the Penguin dictionary prefers to use the terms "belvedere" and
"gazebo" for structures built on top of a roof from which one
can look out. It states that a belfry is the upper room or story of
a tower from which bells are hung. "Lanterns", it says,
are small circular or polygonal turrets with windows all round, crowned
with a roof or dome (probably calling that the cupola).

I myself lean toward the broader definition that the Random House
College Dictionary provides. But I think it could be
embellished a bit. In my opinion, cupolas are
ornamental structures placed in a prominent position, usually at the top
of a larger roof or dome. They often appear as small buildings in
their own right, like diminutive temples perched on top of a
building. They can be the small dome that crowns the top of an
outdoor garden pavilion, folly, or gazebo. Sometimes the pavilion
itself is called a cupola, especially when it takes on the form of a
small, round Roman temple with a dome. Cupolas occasionally act as
the main roof of a tower, spire, or turret. When they do, they
usually provide a considerably more elaborate enclosure than the average
roof.

Cupolas on roofs may be accessible from the inside, commanding a high
vantage point from which to look out over the world. This kind is
often called a belvedere or a "widow's walk". When they
are perforated with windows for illuminating the spaces below, they are
also known as lanterns. This type may or may not be accessible from
the inside. A cupola that maintains a low profile, not poking out
much above the roof, is sometimes called a monitor.

Variations on the word "cupola" appear in other Indo-European
languages, typically referring to a dome. In some of these languages, a
variant of the English word "dome" will indicate a cathedral.
A variant of the English word "lantern" in these same languages
will often come closest in meaning to the word "cupola" in its rooftop
English sense (e.g., Random House's definition 1a).

The Cupola Picture
section of this website has many examples of buildings with cupolas,
domes, lanterns, belvederes, gazebos, and belfries. Some stretch the
definition a bit, but I believe that they warrant inclusion here.
Numerous others may also be seen in the Cupolas
of Capitalism section of this site. I will be adding
additional examples to the cupola galleries as they become available.

For those of you who have been inspired to visit, restore, build, or
buy a cupola, you might want to visit my Cupola
F.A.Q. for more cupola related information and links.